We consider the eigenvalue problem
−
div
(
|
∇
u
|
p
(
x
)
−
2
∇
u
)
=
-\textrm {div}\big (|\nabla u|^{p(x)-2}\nabla u\big )=
λ
V
(
x
)
|
u
|
q
(
x
)
−
2
u
\lambda V(x)|u|^{q(x)-2}u
, in
Ω
\Omega
,
u
=
0
u=0
on
∂
Ω
\partial \Omega
, where
Ω
\Omega
is a smooth bounded domain in
R
N
\mathbb {R}^{N}
,
λ
>
0
\lambda >0
,
p
,
q
p,q
are continuous functions on
Ω
¯
\overline {\Omega }
and
V
V
is a given function in a generalized Lebesgue space
L
s
(
x
)
(
Ω
)
L^{s(x)}(\Omega )
such that
V
>
0
V>0
in an open set
Ω
0
⊂
Ω
\Omega _{0}\subset \Omega
, where
|
Ω
0
|
>
0
|\Omega _{0}| >0
. We prove under appropriate conditions on the functions
p
,
q
p,q
and
s
s
that any
λ
>
0
\lambda >0
sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland’s variational principle.